Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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![### Solving Quadratic Equations by Factoring
This tutorial will guide you through the steps to solve a quadratic equation by factoring. Below is an example problem and its solution process.
**Example Problem:**
Solve the quadratic equation by factoring:
\[ x^2 - 8x - 63 = -10x \]
To solve this equation, follow these steps:
1. **Move All Terms to One Side**:
Start by combining like terms to set the equation equal to zero:
\[ x^2 - 8x - 63 + 10x = 0 \]
Simplify the equation:
\[ x^2 + 2x - 63 = 0 \]
2. **Factor the Quadratic Expression**:
The quadratic expression \( x^2 + 2x - 63 \) needs to be factored. We need to find two numbers that multiply to -63 and add to 2. These numbers are 9 and -7.
Write the expression as:
\[ (x + 9)(x - 7) = 0 \]
3. **Set Each Factor Equal to Zero**:
Set each factor equal to zero and solve for x:
\[ x + 9 = 0 \quad \Rightarrow \quad x = -9 \]
\[ x - 7 = 0 \quad \Rightarrow \quad x = 7 \]
Therefore, the solutions to the quadratic equation \( x^2 + 2x - 63 = 0 \) are:
\[ x = -9 \quad \text{or} \quad x = 7 \]
Once you've worked through the solution, enter your answer in the space provided and submit it to check your result.
**Answer:**
\[ x = \boxed{} \]
[Submit Answer]
By following these steps, you can solve any quadratic equation by factoring. Practice with additional problems to master this fundamental algebra skill.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1fd7f3f7-800b-4386-b3e1-f3ccb9c69bd8%2F0067b28b-8880-48a2-b777-996286691902%2Fe4wju3l_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Solving Quadratic Equations by Factoring
This tutorial will guide you through the steps to solve a quadratic equation by factoring. Below is an example problem and its solution process.
**Example Problem:**
Solve the quadratic equation by factoring:
\[ x^2 - 8x - 63 = -10x \]
To solve this equation, follow these steps:
1. **Move All Terms to One Side**:
Start by combining like terms to set the equation equal to zero:
\[ x^2 - 8x - 63 + 10x = 0 \]
Simplify the equation:
\[ x^2 + 2x - 63 = 0 \]
2. **Factor the Quadratic Expression**:
The quadratic expression \( x^2 + 2x - 63 \) needs to be factored. We need to find two numbers that multiply to -63 and add to 2. These numbers are 9 and -7.
Write the expression as:
\[ (x + 9)(x - 7) = 0 \]
3. **Set Each Factor Equal to Zero**:
Set each factor equal to zero and solve for x:
\[ x + 9 = 0 \quad \Rightarrow \quad x = -9 \]
\[ x - 7 = 0 \quad \Rightarrow \quad x = 7 \]
Therefore, the solutions to the quadratic equation \( x^2 + 2x - 63 = 0 \) are:
\[ x = -9 \quad \text{or} \quad x = 7 \]
Once you've worked through the solution, enter your answer in the space provided and submit it to check your result.
**Answer:**
\[ x = \boxed{} \]
[Submit Answer]
By following these steps, you can solve any quadratic equation by factoring. Practice with additional problems to master this fundamental algebra skill.
![## Solving Quadratics by Factoring: Step-by-Step Example
To solve the quadratic equation by factoring, follow this step-by-step guide.
### Step 1: Set the quadratic expression to zero
Start with your quadratic equation. For this example we have:
\[ x^2 + 14x - 25 = 10x - 4 \]
First, move all the terms to one side to set the equation to zero:
\[ x^2 + 14x - 25 - 10x + 4 = 0 \]
Simplify the expression:
\[ x^2 + 4x - 21 = 0 \]
### Step 2: Factor the quadratic equation
Next, factor the quadratic expression. For this example, the quadratic expression \( x^2 + 4x - 21 \) factors into:
\[ (x + 7)(x - 3) = 0 \]
### Step 3: Set each factor to zero and solve
After factoring the equation, set each factor equal to zero to solve for \( x \):
\[ x + 7 = 0 \quad \text{or} \quad x - 3 = 0 \]
Then, solve each equation:
1. For \( x + 7 = 0 \):
\[ x = -7 \]
2. For \( x - 3 = 0 \):
\[ x = 3 \]
### Solution
The solutions to the quadratic equation \( x^2 + 4x - 21 = 0 \) are:
\[ x = -7 \quad \text{or} \quad x = 3 \]
This step-by-step example illustrates how to solve a quadratic equation by factoring, setting each factor to zero, and solving for \( x \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1fd7f3f7-800b-4386-b3e1-f3ccb9c69bd8%2F0067b28b-8880-48a2-b777-996286691902%2F2nhjnll_processed.jpeg&w=3840&q=75)
Transcribed Image Text:## Solving Quadratics by Factoring: Step-by-Step Example
To solve the quadratic equation by factoring, follow this step-by-step guide.
### Step 1: Set the quadratic expression to zero
Start with your quadratic equation. For this example we have:
\[ x^2 + 14x - 25 = 10x - 4 \]
First, move all the terms to one side to set the equation to zero:
\[ x^2 + 14x - 25 - 10x + 4 = 0 \]
Simplify the expression:
\[ x^2 + 4x - 21 = 0 \]
### Step 2: Factor the quadratic equation
Next, factor the quadratic expression. For this example, the quadratic expression \( x^2 + 4x - 21 \) factors into:
\[ (x + 7)(x - 3) = 0 \]
### Step 3: Set each factor to zero and solve
After factoring the equation, set each factor equal to zero to solve for \( x \):
\[ x + 7 = 0 \quad \text{or} \quad x - 3 = 0 \]
Then, solve each equation:
1. For \( x + 7 = 0 \):
\[ x = -7 \]
2. For \( x - 3 = 0 \):
\[ x = 3 \]
### Solution
The solutions to the quadratic equation \( x^2 + 4x - 21 = 0 \) are:
\[ x = -7 \quad \text{or} \quad x = 3 \]
This step-by-step example illustrates how to solve a quadratic equation by factoring, setting each factor to zero, and solving for \( x \).
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